When you live in a first-world, developed country it's pretty easy to isolate yourself from the reality of life for the majority of the world, so first educate yourself. You can read books about the subject, learn facts about poverty and see if for yourself.
Once you've educated yourself, it's fairly easy to share what you've learned with others. If you read a good book about the subject, pass that book on to a friend and encourage them to read it. As you learn facts, share them with others..."Did you know....?" If you are able to visit a third world country, take lots of photos and collect stories to share with your family and friends when you return.
I've been learning that people are much more willing to listen if you aren't trying to convince them to do something. When you share from your heart about what God is teaching you, others will be drawn to your passion. It's not our job to convince others to be interested in what we are interested in. We are called to plant the seeds and share the truth.
Not everyone will choose to fight poverty through child sponsorship. That's ok. There are many, many, many good ministries out there that work in different ways. Just as you want to be supported in your advocacy for child sponsorship, support others and show an interest in what they are passionate about. If a friend or family member listens to what you have to share about your sponsorship experience, ask them about the ministries and missions they support.
It's not a contest to see how many people we can recruit to our side. It's about mutual respect and giving one another encouragement. Your family and friends don't have to sponsor a child to support the work you are doing. Sponsorship is one way (a very, very good way!) to fight poverty, but there are other ways.
I think you'll find so much freedom when you drop the expectation that you are supposed to convince everyone else to sponsor a child.
Be authentic and share with integrity. There is a huge problem in the world and it's called extreme poverty. You know it exists and you know God has given you a role in eliminating it. You've read the Scriptures and it's clear you are to respond. You've found a ministry that is committed to Christ, the local church, children and integrity and you trust them to continue releasing children from poverty in Jesus' name. That's all you need to share.